On the Solution of Fractional Differential-Algebraic Systems With the Adomian Decomposition Method
-
摘要: 研究了利用Adomian分解求解分数阶微分代数系统的方法.分析了代数约束对Adomian方法求解的影响,指出直接解出代数约束变量,将原系统转化为微分系统进行Adomian分解的困难.提出确定代数变量级数解各分量的新方法,据此进行Adomian分解,得到整个系统的级数解.特别研究了代数约束为线性的分数阶微分代数系统的Adomian解法,证明了各变量间的线性代数约束关系可以转化为相应级数解中各分量的线性关系,从而方便求解,并结合具体例子证明了该方法简便有效.Abstract: The solution of the fractional differentialalgebraic systems (FDASs) was studied with the Adomian decomposition method. The influence of the algebraic constraints on the Adomian decomposition method was investigated, and the main difficulty of transforming the FDASs into fractional differential systems through solving the algebraic constraints directly was pointed out. To determine the components of the algebraic variable series solution, a new method was presented with the Adomian decomposition implemented successfully to obtain the solution of the FDAS. The solution of the FDAS under linear algebraic constraints was particularly discussed with the Adomian decomposition method. It’s proved that the linear relationship between the variables under algebraic constraints could be equivalently transformed into the linear relationship between the components of the corresponding series solution. 2 examples were given to illustrate the convenience and effectiveness of the proposed method.
-
[1] Zurigat M, Momani S. Analytical approximate solutions of systems of fractional algebraic-differential equations by homotopy analysis method[J].Computers & Mathematics With Applications,2010,59(3): 1227-1235. [2] DING Xiao-li, JIANG Yao-lin. Waveform relaxation method for fractional differential-algebraic equations[J].Fractional Calculus &Applied Analysis,2014,17(3): 585-604. [3] Adomian G, Rach R. Inversion of nonlinear stochastic operators[J].Journal of Mathematical Analysis and Applications, 1983,91(1): 39-46. [4] Adomian G. Stochastic Systems [M]. New York: Academic Press, 1983. [5] Bellman R E, Adomian G.Partial Differential Equations: New Methods for Their Treatment and Solution[M]. Dordrecht: D Reidel Publishing Company, 1985. [6] Adomian G.Nonlinear Stochastic Operator Equations [M]. San Diego: Academic Press, 1986. [7] Bellomo N, Riganti R.Nonlinear Stochastic Systems Analysis in Physics and Mechanics [M]. Singapore: World Scientific Publishing Company, 1987. [8] Adomian G.Nonlinear Stochastic Systems Theory and Applications to Physics [M]. Dordrecht: Kluwer Academic, 1989. [9] Arora H L, Abdelwahid F I. Solution of non-integer order differential equations via the Adomian decomposition method[J].Applied Mathematics Letters,1993,6(1): 21-23. [10] George A J, Chakrabarti A. The Adomian method applied to some extraordinary differential equations[J].Applied Mathematics Letters, 1995,8(3): 391-397. [11] Shawagfeh N T. Analytical approximate solutions for nonlinear fractional differential equations[J].Applied Mathematics and Computation,2002,131(2): 517-529. [12] Daftardar-Gejji V, Jafari H. Adomian decomposition: a tool for solving a system of fractional differential equations[J].Journal of Mathematical Analysis and Applications,2005,301(2): 508-518. [13] Jafari H, Daftardar-Gejji V. Solving a system of nonlinear fractional differential equations using Adomian decomposition[J].Journal of Computational & Applied Mathematics, 2006,196(2): 644-651. [14] LI Chang-pin, WANG Yi-hong. Numerical algorithm based on Adomian decomposition for fractional differential equations[J].Computers & Mathematics With Applications,2009,57(10): 1672-1681. [15] Duan J S. Recurrence triangle for Adomian polynomials[J].Applied Mathematics and Computation,2010,216(4): 1235-1241. [16] Duan J S, Rach R, Wang Z. On the effective region of convergence of the decomposition series solution[J].Journal of Algorithms & Computational Technology,2013,2: 227-248.
点击查看大图
计量
- 文章访问数: 1434
- HTML全文浏览量: 80
- PDF下载量: 713
- 被引次数: 0